Q. 80 Prove聽that聽if聽f聽is聽continuo... [FREE SOLUTION]

Chapter 6: Q. 80 (page 541)

$Prove that if f is continuous on [a, b] and C is any real number, then f (x) and f (x) + C have the same arc length on [a, b] . Then explain why this makes sense graphically .$

Short Answer

Expert verified

$The arc length of f (x) from x = a to x = b can be represented by the definite integral : {鈭�}_{a}^{b} \sqrt{1 + (f' {(x))}^{2}} d x Here, when f (x) and f (x) + C are differentiated, we get the same f' (x) as differentiation of a constant is zero . when this is substituted in the definite integral we get the same arc length in the same interval . Graphically this makes sense as the f (x) + C is vertically shifted version of f (x) and arc lengths of both the curves are same in the same interval .$

Step by step solution

Step 1. Given information

$f (x) is continuous on [a, b] and C is any real number$ .

Step 2. Finding arc lengths of f(x) and f(x)+C and proving why both are same.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

$Prove that if f is continuous on [a, b] and C is any real number, then f (x) and f (x) + C have the same arc length on [a, b] . Then explain why this makes sense graphically .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding arc lengths of f(x) and f(x)+C and proving why both are same.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

Provethatiffiscontinuouson[a,b]andCisanyrealnumber,thenf(x)andf(x)+Chavethesamearclengthon[a,b].Thenexplainwhythismakessensegraphically.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding arc lengths of f(x) and f(x)+C and proving why both are same.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

$Prove that if f is continuous on [a, b] and C is any real number, then f (x) and f (x) + C have the same arc length on [a, b] . Then explain why this makes sense graphically .$